We show that an n x n complex matrix T is the product of two unipotent matrices of index 2 if and only if T is similar to a matrix of the form D + D-1 + (I + N) + (- I + SIGMA-i(m) = 1 + J(i)), where 0 and +/- 1 are not eigenvalues of D, N is nilpotent, and each J(i) is a nilpotent Jordan block of even size. On the other hand, T is the product of finitely many unipotent matrices of index 2 if and only if det T = 1. In this case, the minimal number of required unipotents is 1 if n = 1, 3 if n = 2, and 4 if n greater-than-or-equal-to 3.